In my lectures, I will describe a broad program that has been initiated to model information flow in these and related areas. This includes a high-level reformulation of quantum information and quantum computing using category theory that have been shown to capture all of the fundamental components of the theory. The approach supports reasoning about classical and quantum communication in the same model. The approach also has provided what are arguably the first completely formal descriptions and proofs of correctness of several key quantum informatic protocols, e.g. (logic-gate) teleportation, superdense coding, and one-way computational schemes. It also provides a description of the quantum state, as well as the flow of information from the quantum state to the classical world (measurements), and from the classical world to the quantum state (control), all of which are important for reasoning about security in a quantum setting.' http://129.81.170.14/~mwm/clifford/Site/Abstracts.html

4.
ThemesIntroduction• Themes • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spaces• Outline I Exempliﬁes one of the main thrusts of our group in Oxford:• Outline II methods and concepts which have been developed in TheoreticalChu Spaces Computer Science are ripe for use in Physics.Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3

5.
ThemesIntroduction• Themes • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spaces• Outline I Exempliﬁes one of the main thrusts of our group in Oxford:• Outline II methods and concepts which have been developed in TheoreticalChu Spaces Computer Science are ripe for use in Physics.Representing PhysicalSystemsCharacterizing Chu • Models vs. Axioms. Examples: sheaves and toposes,Morphisms on domain-theoretic models of the λ-calculus.Quantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3

6.
ThemesIntroduction• Themes • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spaces• Outline I Exempliﬁes one of the main thrusts of our group in Oxford:• Outline II methods and concepts which have been developed in TheoreticalChu Spaces Computer Science are ripe for use in Physics.Representing PhysicalSystemsCharacterizing Chu • Models vs. Axioms. Examples: sheaves and toposes,Morphisms on domain-theoretic models of the λ-calculus.Quantum Chu SpacesThe RepresentationTheorem • Toy Models. Rob Spekkens: ‘Evidence for the epistemic view ofReducing The Value quantum states: A toy theory’.SetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3

7.
ThemesIntroduction• Themes • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spaces• Outline I Exempliﬁes one of the main thrusts of our group in Oxford:• Outline II methods and concepts which have been developed in TheoreticalChu Spaces Computer Science are ripe for use in Physics.Representing PhysicalSystemsCharacterizing Chu • Models vs. Axioms. Examples: sheaves and toposes,Morphisms on domain-theoretic models of the λ-calculus.Quantum Chu SpacesThe RepresentationTheorem • Toy Models. Rob Spekkens: ‘Evidence for the epistemic view ofReducing The Value quantum states: A toy theory’.SetDiscussionChu Spaces and • Big toy models.CoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3

9.
Chu SpacesIntroduction We should understand Chu spaces as providing a very general (and, we• Themes• Chu Spaces might reasonably say, rather simple) ‘logic of systems or structures’.• Outline I• Outline II Indeed, they have been proposed by Barwise and Seligman as theChu Spaces vehicle for a general logic of ‘distributed systems’ and information ﬂow.Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 4

10.
Chu SpacesIntroduction We should understand Chu spaces as providing a very general (and, we• Themes• Chu Spaces might reasonably say, rather simple) ‘logic of systems or structures’.• Outline I• Outline II Indeed, they have been proposed by Barwise and Seligman as theChu Spaces vehicle for a general logic of ‘distributed systems’ and information ﬂow.Representing PhysicalSystems This logic of Chu spaces was in no way biassed in its conception towardsCharacterizing Chu the description of quantum mechanics or any other kind of physicalMorphisms onQuantum Chu Spaces system.The RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 4

11.
Chu SpacesIntroduction We should understand Chu spaces as providing a very general (and, we• Themes• Chu Spaces might reasonably say, rather simple) ‘logic of systems or structures’.• Outline I• Outline II Indeed, they have been proposed by Barwise and Seligman as theChu Spaces vehicle for a general logic of ‘distributed systems’ and information ﬂow.Representing PhysicalSystems This logic of Chu spaces was in no way biassed in its conception towardsCharacterizing Chu the description of quantum mechanics or any other kind of physicalMorphisms onQuantum Chu Spaces system.The RepresentationTheorem Just for this reason, it is interesting to see how much ofReducing The ValueSet quantum-mechanical structure and concepts can be absorbed andDiscussion essentially determined by this more general systems logic.Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 4

19.
Outline II • This leads to a further question of conceptual interest: is this representation preserved by collapsing the unit interval to ﬁnitely many values?Big Toy Models Workshop on Informatic Penomena 2009 – 6

20.
Outline II • This leads to a further question of conceptual interest: is this representation preserved by collapsing the unit interval to ﬁnitely many values? • For the two canonical possibilistic collapses to two values, we show that this fails.Big Toy Models Workshop on Informatic Penomena 2009 – 6

21.
Outline II • This leads to a further question of conceptual interest: is this representation preserved by collapsing the unit interval to ﬁnitely many values? • For the two canonical possibilistic collapses to two values, we show that this fails. • However, the natural collapse to three values works! — A possible role ˆ for 3-valued logic in quantum foundations?Big Toy Models Workshop on Informatic Penomena 2009 – 6

22.
Outline II • This leads to a further question of conceptual interest: is this representation preserved by collapsing the unit interval to ﬁnitely many values? • For the two canonical possibilistic collapses to two values, we show that this fails. • However, the natural collapse to three values works! — A possible role ˆ for 3-valued logic in quantum foundations? • We also look at coalgebras as a possible alternative setting to Chu spaces. Some interesting and novel points arise in comparing and relating these two well-studied systems models.Big Toy Models Workshop on Informatic Penomena 2009 – 6

23.
Outline II • This leads to a further question of conceptual interest: is this representation preserved by collapsing the unit interval to ﬁnitely many values? • For the two canonical possibilistic collapses to two values, we show that this fails. • However, the natural collapse to three values works! — A possible role ˆ for 3-valued logic in quantum foundations? • We also look at coalgebras as a possible alternative setting to Chu spaces. Some interesting and novel points arise in comparing and relating these two well-studied systems models. There is a paper available as an Oxford University Computing Laboratory Research Report: RR–09–08 at http://www.comlab.ox.ac.uk/techreports/cs/2009.htmlBig Toy Models Workshop on Informatic Penomena 2009 – 6

29.
Chu Spaces History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by Po-Hsiang Chu. A generalization of constructions of dual pairings of topological vector spaces from G. W. Mackey’s thesis. Chu spaces have several interesting aspects: • They have a rich type structure, and in particular form models of Linear Logic (Seely). • They have a rich representation theory; many concrete categories of interest can be fully embedded into Chu spaces (Lafont and Streicher, Pratt).Big Toy Models Workshop on Informatic Penomena 2009 – 8

30.
Chu Spaces History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by Po-Hsiang Chu. A generalization of constructions of dual pairings of topological vector spaces from G. W. Mackey’s thesis. Chu spaces have several interesting aspects: • They have a rich type structure, and in particular form models of Linear Logic (Seely). • They have a rich representation theory; many concrete categories of interest can be fully embedded into Chu spaces (Lafont and Streicher, Pratt). • There is a natural notion of ‘local logic’ on Chu spaces (Barwise), and an interesting characterization of information transfer across Chu morphisms (van Benthem).Big Toy Models Workshop on Informatic Penomena 2009 – 8

31.
Chu Spaces History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by Po-Hsiang Chu. A generalization of constructions of dual pairings of topological vector spaces from G. W. Mackey’s thesis. Chu spaces have several interesting aspects: • They have a rich type structure, and in particular form models of Linear Logic (Seely). • They have a rich representation theory; many concrete categories of interest can be fully embedded into Chu spaces (Lafont and Streicher, Pratt). • There is a natural notion of ‘local logic’ on Chu spaces (Barwise), and an interesting characterization of information transfer across Chu morphisms (van Benthem). Applications of Chu spaces have been proposed in a number of areas, including concurrency, hardware veriﬁcation, game theory and fuzzy systems.Big Toy Models Workshop on Informatic Penomena 2009 – 8

33.
Deﬁnitions Fix a set K . A Chu space over K is a structure (X, A, e), where X is a set of ‘points’ or ‘objects’, A is a set of ‘attributes’, and e : X × A → K is an evaluation function.Big Toy Models Workshop on Informatic Penomena 2009 – 9

49.
The General Paradigm We take a system to be speciﬁed by its set of states S , and the set of questions Q which can be ‘asked’ of the system.Big Toy Models Workshop on Informatic Penomena 2009 – 13

50.
The General Paradigm We take a system to be speciﬁed by its set of states S , and the set of questions Q which can be ‘asked’ of the system. We shall consider only ‘yes/no’ questions; however, the result of asking a question in a given state will in general be probabilistic. This will be represented by an evaluation function e : S × Q → [0, 1] where e(s, q) is the probability that the question q will receive the answer ‘yes’ when the system is in state s.Big Toy Models Workshop on Informatic Penomena 2009 – 13

51.
The General Paradigm We take a system to be speciﬁed by its set of states S , and the set of questions Q which can be ‘asked’ of the system. We shall consider only ‘yes/no’ questions; however, the result of asking a question in a given state will in general be probabilistic. This will be represented by an evaluation function e : S × Q → [0, 1] where e(s, q) is the probability that the question q will receive the answer ‘yes’ when the system is in state s. This is a Chu space!Big Toy Models Workshop on Informatic Penomena 2009 – 13

52.
The General Paradigm We take a system to be speciﬁed by its set of states S , and the set of questions Q which can be ‘asked’ of the system. We shall consider only ‘yes/no’ questions; however, the result of asking a question in a given state will in general be probabilistic. This will be represented by an evaluation function e : S × Q → [0, 1] where e(s, q) is the probability that the question q will receive the answer ‘yes’ when the system is in state s. This is a Chu space! N.B. This is essentially the point of view taken by Mackey in his classic ‘Mathematical foundations of Quantum Mechanics’. Note that we prefer ‘question’ to ‘property’, since QM we cannot think in terms of static properties which are determinately possessed by a given state; questions imply a dynamic act of asking.Big Toy Models Workshop on Informatic Penomena 2009 – 13

53.
The General Paradigm We take a system to be speciﬁed by its set of states S , and the set of questions Q which can be ‘asked’ of the system. We shall consider only ‘yes/no’ questions; however, the result of asking a question in a given state will in general be probabilistic. This will be represented by an evaluation function e : S × Q → [0, 1] where e(s, q) is the probability that the question q will receive the answer ‘yes’ when the system is in state s. This is a Chu space! N.B. This is essentially the point of view taken by Mackey in his classic ‘Mathematical foundations of Quantum Mechanics’. Note that we prefer ‘question’ to ‘property’, since QM we cannot think in terms of static properties which are determinately possessed by a given state; questions imply a dynamic act of asking. It is standard in the foundational literature on QM to focus on yes/no questions. However, the usual approaches to quantum logic avoid the direct introduction of probabilities. More on this later!Big Toy Models Workshop on Informatic Penomena 2009 – 13

56.
Representing Quantum Systems As Chu Spaces A quantum system with a Hilbert space H as its state space will beIntroduction represented asChu Spaces (H◦ , L(H), eH )Representing PhysicalSystems• The General whereParadigm• RepresentingQuantum Systems As • H◦ is the set of non-zero vectors of HChu SpacesCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14

57.
Representing Quantum Systems As Chu Spaces A quantum system with a Hilbert space H as its state space will beIntroduction represented asChu Spaces (H◦ , L(H), eH )Representing PhysicalSystems• The General whereParadigm• RepresentingQuantum Systems As • H◦ is the set of non-zero vectors of HChu SpacesCharacterizing ChuMorphisms on • L(H) is the set of closed subspaces of H — the ‘yes/no’ questionsQuantum Chu Spaces of QMThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14

62.
OverviewIntroduction We shall now see how the simple, discrete notions of Chu spaces sufﬁceChu Spaces to determine the appropriate notions of state equivalence, and to pick outRepresenting PhysicalSystems the physically signiﬁcant symmetries on Hilbert space in a very strikingCharacterizing Chu fashion. This leads to a full and faithful representation of the category ofMorphisms onQuantum Chu Spaces quantum systems, with the groupoid structure of their physical• Overview• Biextensionaity symmetries, in the category of Chu spaces valued in the unit interval.• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 16

63.
OverviewIntroduction We shall now see how the simple, discrete notions of Chu spaces sufﬁceChu Spaces to determine the appropriate notions of state equivalence, and to pick outRepresenting PhysicalSystems the physically signiﬁcant symmetries on Hilbert space in a very strikingCharacterizing Chu fashion. This leads to a full and faithful representation of the category ofMorphisms onQuantum Chu Spaces quantum systems, with the groupoid structure of their physical• Overview• Biextensionaity symmetries, in the category of Chu spaces valued in the unit interval.• Projectivity =Biextensionality• Characterizing Chu The arguments here make use of Wigner’s theorem and the dualities ofMorphisms• Injectivity projective geometry, in the modern form developed by Faure andAssumption ¨ Frolicher, Modern Projective Geometry (2000).• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 16

64.
OverviewIntroduction We shall now see how the simple, discrete notions of Chu spaces sufﬁceChu Spaces to determine the appropriate notions of state equivalence, and to pick outRepresenting PhysicalSystems the physically signiﬁcant symmetries on Hilbert space in a very strikingCharacterizing Chu fashion. This leads to a full and faithful representation of the category ofMorphisms onQuantum Chu Spaces quantum systems, with the groupoid structure of their physical• Overview• Biextensionaity symmetries, in the category of Chu spaces valued in the unit interval.• Projectivity =Biextensionality• Characterizing Chu The arguments here make use of Wigner’s theorem and the dualities ofMorphisms• Injectivity projective geometry, in the modern form developed by Faure andAssumption ¨ Frolicher, Modern Projective Geometry (2000).• Orthogonality isPreserved• Constructing the LeftAdjoint The surprising point is that unitarity/anitunitarity is essentially forced by• Using Projective the mere requirement of being a Chu morphism. This even extends toDuality• Wigner’s Theorem surjectivity, which here is derived rather than assumed.• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 16

89.
Using Projective Duality By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a left adjoint, it preserves sups. Hence f → and f∗ are paired under the duality of projective lattices and projective geometries.Big Toy Models Workshop on Informatic Penomena 2009 – 23

90.
Using Projective Duality By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a left adjoint, it preserves sups. Hence f → and f∗ are paired under the duality of projective lattices and projective geometries. Proposition 11 f∗ is a total map of projective geometries.Big Toy Models Workshop on Informatic Penomena 2009 – 23

91.
Using Projective Duality By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a left adjoint, it preserves sups. Hence f → and f∗ are paired under the duality of projective lattices and projective geometries. Proposition 11 f∗ is a total map of projective geometries. We can now apply Wigner’s Theorem, in the modernized form given by Faure (2002).Big Toy Models Workshop on Informatic Penomena 2009 – 23

92.
Using Projective Duality By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a left adjoint, it preserves sups. Hence f → and f∗ are paired under the duality of projective lattices and projective geometries. Proposition 11 f∗ is a total map of projective geometries. We can now apply Wigner’s Theorem, in the modernized form given by Faure (2002). Let V1 be a vector space over the ﬁeld F and V2 a vector space over the ﬁeld G. A semilinear map from V1 to V2 is a pair (f, α) where α : F → G is a ﬁeld homomorphism, and f : V1 → V2 is an additive map such that, for all λ ∈ F and v ∈ V1 : f (λv) = α(λ)f (v).Big Toy Models Workshop on Informatic Penomena 2009 – 23

93.
Using Projective Duality By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a left adjoint, it preserves sups. Hence f → and f∗ are paired under the duality of projective lattices and projective geometries. Proposition 11 f∗ is a total map of projective geometries. We can now apply Wigner’s Theorem, in the modernized form given by Faure (2002). Let V1 be a vector space over the ﬁeld F and V2 a vector space over the ﬁeld G. A semilinear map from V1 to V2 is a pair (f, α) where α : F → G is a ﬁeld homomorphism, and f : V1 → V2 is an additive map such that, for all λ ∈ F and v ∈ V1 : f (λv) = α(λ)f (v). Note that semilinear maps compose: if (f, α) : V1 → V2 and (g, β) : V2 → V3 , then (g ◦ f, β ◦ α) : V1 → V2 is a semilinear map.Big Toy Models Workshop on Informatic Penomena 2009 – 23

94.
Using Projective Duality By construction, f → extends f∗ : this says that f → preserves atoms. Since f → is a left adjoint, it preserves sups. Hence f → and f∗ are paired under the duality of projective lattices and projective geometries. Proposition 11 f∗ is a total map of projective geometries. We can now apply Wigner’s Theorem, in the modernized form given by Faure (2002). Let V1 be a vector space over the ﬁeld F and V2 a vector space over the ﬁeld G. A semilinear map from V1 to V2 is a pair (f, α) where α : F → G is a ﬁeld homomorphism, and f : V1 → V2 is an additive map such that, for all λ ∈ F and v ∈ V1 : f (λv) = α(λ)f (v). Note that semilinear maps compose: if (f, α) : V1 → V2 and (g, β) : V2 → V3 , then (g ◦ f, β ◦ α) : V1 → V2 is a semilinear map. N.B. There are lots of (horrible) automorphisms, and non-surjective endomorphisms, of the complex ﬁeld!Big Toy Models Workshop on Informatic Penomena 2009 – 23

104.
A Surprise: Surjectivity Comes for Free! Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗ where f is a Chu space morphism, and dim(H) 0. Suppose that the endomorphism σ : C → C associated with g is surjective, and hence an automorphism. Then g is surjective. Proof We write Im g for the set-theoretic direct image of g , which is a linear subspace of K, since σ is an automorphism. In particular, g carries rays to rays, since λg(φ) = g(σ −1 (λ)φ).Big Toy Models Workshop on Informatic Penomena 2009 – 26

105.
A Surprise: Surjectivity Comes for Free! Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗ where f is a Chu space morphism, and dim(H) 0. Suppose that the endomorphism σ : C → C associated with g is surjective, and hence an automorphism. Then g is surjective. Proof We write Im g for the set-theoretic direct image of g , which is a linear subspace of K, since σ is an automorphism. In particular, g carries rays to rays, since λg(φ) = g(σ −1 (λ)φ). We claim that for any vector ψ ∈ K◦ which is not in the image of g , ψ ⊥ Im g . ¯ ¯ Given such a ψ , for any φ ∈ H◦ it is not the case that f∗ (φ) ⊆ ψ ; for otherwise, for some λ, g(φ) = λψ , and hence g(σ −1 (λ−1 )φ) = ψ . Then by Proposition 6, ¯ f ∗ (ψ) = {0}. It follows that for all φ ∈ H◦ , ¯ ¯ ¯ ¯ eK (f∗ (φ), ψ) = eH (φ, {0}) = 0, ¯ and hence by Lemma 8 that ψ ⊥ Im g .Big Toy Models Workshop on Informatic Penomena 2009 – 26

106.
A Surprise: Surjectivity Comes for Free! Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗ where f is a Chu space morphism, and dim(H) 0. Suppose that the endomorphism σ : C → C associated with g is surjective, and hence an automorphism. Then g is surjective. Proof We write Im g for the set-theoretic direct image of g , which is a linear subspace of K, since σ is an automorphism. In particular, g carries rays to rays, since λg(φ) = g(σ −1 (λ)φ). We claim that for any vector ψ ∈ K◦ which is not in the image of g , ψ ⊥ Im g . ¯ ¯ Given such a ψ , for any φ ∈ H◦ it is not the case that f∗ (φ) ⊆ ψ ; for otherwise, for some λ, g(φ) = λψ , and hence g(σ −1 (λ−1 )φ) = ψ . Then by Proposition 6, ¯ f ∗ (ψ) = {0}. It follows that for all φ ∈ H◦ , ¯ ¯ ¯ ¯ eK (f∗ (φ), ψ) = eH (φ, {0}) = 0, ¯ and hence by Lemma 8 that ψ ⊥ Im g . Now suppose for a contradiction that such a ψ exists. Consider the vector ψ + χ where χ is a non-zero vector in Im g , which must exist since g is injective and H has positive dimension. This vector is not in Im g , nor is it orthogonal to Im g , since e.g. ψ + χ | χ = χ | χ = 0. This yields the required contradiction.Big Toy Models Workshop on Informatic Penomena 2009 – 26

115.
The Big Picture We deﬁne a category SymmH as follows: • The objects are Hilbert spaces of dimension 2. • Morphisms U : H → K are semiunitary (i.e. unitary or antiunitary) maps. • Semiunitaries compose as explained more generally for semilinear maps in the previous subsection. Since complex conjugation is an involution, semiunitaries compose according to the rule of signs: two antiunitaries or two unitaries compose to form a unitary, while a unitary and an antiunitary compose to form an antiunitary.Big Toy Models Workshop on Informatic Penomena 2009 – 29

116.
The Big Picture We deﬁne a category SymmH as follows: • The objects are Hilbert spaces of dimension 2. • Morphisms U : H → K are semiunitary (i.e. unitary or antiunitary) maps. • Semiunitaries compose as explained more generally for semilinear maps in the previous subsection. Since complex conjugation is an involution, semiunitaries compose according to the rule of signs: two antiunitaries or two unitaries compose to form a unitary, while a unitary and an antiunitary compose to form an antiunitary. This category is a groupoid, i.e. every arrow is an isomorphism.Big Toy Models Workshop on Informatic Penomena 2009 – 29

117.
The Big Picture We deﬁne a category SymmH as follows: • The objects are Hilbert spaces of dimension 2. • Morphisms U : H → K are semiunitary (i.e. unitary or antiunitary) maps. • Semiunitaries compose as explained more generally for semilinear maps in the previous subsection. Since complex conjugation is an involution, semiunitaries compose according to the rule of signs: two antiunitaries or two unitaries compose to form a unitary, while a unitary and an antiunitary compose to form an antiunitary. This category is a groupoid, i.e. every arrow is an isomorphism. The seminunitaries are the physically signiﬁcant symmetries of Hilbert space from the point of view of Quantum Mechanics. The usual dynamics according to the Schrodinger equation is given by a continuous one-parameter group {U (t)} of ¨ these symmetries; the requirement of continuity forces the U (t) to be unitaries. However, some important physical symmetries are represented by antiunitaries, e.g. time reversal and charge conjugation.Big Toy Models Workshop on Informatic Penomena 2009 – 29

119.
Remarks • By the results of the previous subsection, Chu morphisms essentially force us to consider the symmetries on Hilbert space. As pointed out there, linear maps which can be represented as Chu morphisms in the biextensional category must be injective; and if L : H → K is an injective linear or antilinear map, then P(L) is injective.Big Toy Models Workshop on Informatic Penomena 2009 – 30

120.
Remarks • By the results of the previous subsection, Chu morphisms essentially force us to consider the symmetries on Hilbert space. As pointed out there, linear maps which can be represented as Chu morphisms in the biextensional category must be injective; and if L : H → K is an injective linear or antilinear map, then P(L) is injective. • Our results then show that if L can be represented as a Chu morphism, it must in fact be semiunitary.Big Toy Models Workshop on Informatic Penomena 2009 – 30